Square Root of 1058

The square root is the inverse of the square of the number. 1058 is not a perfect square. The square root of 1058 is expressed in both radical and exponential form. In the radical form, it is expressed as √1058, whereas (1058)^(1/2) in the exponential form. √1058 ≈ 32.529, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 1058 is broken down into its prime factors:

Step 1: Finding the prime factors of 1058 Breaking it down, we get 2 x 23 x 23: 2^1 x 23^2

Step 2: Now we found out the prime factors of 1058. Since 1058 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1058 using prime factorization does not yield a whole number square root.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: To begin with, we need to group the numbers from right to left. In the case of 1058, we need to group it as 58 and 10.

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n is ‘3’ because 3 × 3 = 9, which is less than 10. Now the quotient is 3, after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 58, making the new dividend 158. Add the old divisor with the same number 3 + 3 = 6, which will be part of our new divisor.

Step 4: The new divisor is 6n, and we need to find the value of n such that 6n × n ≤ 158. Let us consider n as 2, now 62 × 2 = 124.

Step 5: Subtracting 124 from 158 gives a remainder of 34, and the quotient is 32.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 3400.

Step 7: Now, continue the process to find decimal places.

The square root of 1058 is approximately 32.529.

Approximation method is another method for finding the square roots, and it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1058 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √1058. The smallest perfect square less than 1058 is 1024 (32^2), and the largest perfect square greater than 1058 is 1089 (33^2). √1058 falls somewhere between 32 and 33.

Step 2: Now we approximate between these values. Since 1058 is closer to 1024, the square root will be closer to 32. Using a calculator or further steps.

We find the square root of 1058 is approximately 32.529.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A perfect square is a number that can be expressed as the product of an integer with itself. Example: 36 is a perfect square because 6 x 6 = 36.
   
• Long division method: A method used to find the square root of non-perfect square numbers through successive approximation and division.
   
• Approximation method: A method used to estimate the square root of a number by identifying the closest perfect squares and estimating between them.